It says the Spaceship One carrier uses J85s to climb to 53,000 ft. Did they choose turbojets (instead of fans) because of better high-altitude performance? If so, is it just the afterburner that makes it perform better, or does the J85 itself outperform a comparable fan?

I believe the J85-GE-5's were chosen because they were available and they could get them at a very low price. A comparable turbofan would be something like A TF-731 or a PWC PW-300, both vastly more expensive engines even on the used market, and much heavier as well. I do remember watching the documentary and seeing an F-5 or T-38 trainer in the background sans engines, so I it's possible they were removed to that aircraft and transplanted to White Knight. The all up weight for White Knight and Spaceship One was about 17,000lbs giving a thrust to weight ratio of 2.2:1, and without Spaceship One close to unity. The J85's thrust would be behind those turbofans I mentioned at altitudes below 10,000 feet or so but after that the J85 would leave them far behind as far as thrust is concerned.